Problem 88 - Algebra and Trigonometry

Question

Simplify the expression. (This type of expression arises in calculus when using the “quotient rule.”) $$ \frac{(7-3 x)^{1 / 2}+\frac{3}{2} x(7-3 x)^{-1 / 2}}{7-3 x} $$

Step-by-Step Solution

Answer

The simplified expression is \((7-\frac{3}{2}x)(7-3x)^{-3/2}\).

Step 1: Simplify the Numerator

The numerator of the expression is \((7-3x)^{1/2} + \frac{3}{2}x(7-3x)^{-1/2}\). Notice that both terms in the numerator have a common factor that involves \((7-3x)\). Write the first term as a single fraction: \(\frac{(7-3x)^{1/2} \cdot (7-3x)}{(7-3x)} = \frac{(7-3x)^{1/2}}{1}\). The second term is already a fraction: \(\frac{3}{2}x(7-3x)^{-1/2}\). Identify the common factor \((7-3x)^{-1/2}\) and reorganize if needed.

Step 2: Factor and Combine

Factor out \((7-3x)^{-1/2}\) from the terms in the numerator: \[(7-3x)^{-1/2} \left( (7-3x) + \frac{3}{2}x \right)\] Now, combine the terms inside the parentheses. This gives: \[(7-3x)^{-1/2} \left( 7 - 3x + \frac{3}{2}x \right)\]This simplifies to:\[(7-3x)^{-1/2} \left( 7 - \frac{3}{2}x \right)\]

Step 3: Simplify the Division

The expression for the entire fraction is: \[\frac{(7-3x)^{-1/2} \left( 7 - \frac{3}{2}x \right)}{7-3x}\]This can be rewritten as: \[\left(7-\frac{3}{2}x\right) \cdot (7-3x)^{-3/2}\] Thus, the expression is simplified.

Key Concepts

Understanding the Quotient Rule

The quotient rule is a fundamental tool used in calculus to differentiate functions that are presented as a ratio of two other functions. It allows us to find the derivative of such expressions. Although primarily used in calculus, the concept can be helpful when simplifying expressions, as it involves careful handling of numerators and denominators.

When using the quotient rule, the derivative of a quotient \( \frac{u(x)}{v(x)} \) is given by:

• \( \left( \frac{d}{dx} \right) \frac{u(x)}{v(x)} = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \)

In our exercise, the initial expression is a complex combination of terms with both bases and exponents involved. While not directly a calculus problem, simplifying such expressions can involve similar techniques. You treat the entire numerator and denominator to maximize simplification, using common factors.
This approach helps to streamline expressions for further mathematical manipulation or evaluation.

Principles of Factorization

Factorization is a process of breaking down expressions into products of their simpler parts, or 'factors'. It's a vital tool in simplifying algebraic expressions as it allows us to identify and cancel out common components.

In the given expression, factorization helps to identify common terms like \((7-3x)\) found in both the numerator and denominator. Here is how factorization is applied:

• Determine the greatest common factor among terms, such as \((7-3x)^{-1/2}\) in the numerator of the problem.
• Factor this common term out of the numerator, simplifying the expression inside it.
• Be cautious: all terms inside the factor brace must be handled correctly, respecting their coefficients.

By thoughtfully applying factorization principles, expressions can be manipulated into simpler forms that are easier to work with, ultimately leading to a more polished and elegant solution.

Algebraic Fractions Simplification

Algebraic fractions are similar to numerical fractions, but they contain algebraic expressions instead of numbers. Simplifying them involves reducing the complexity of these expressions by utilizing techniques such as factoring and finding common denominators.

Here's how to handle algebraic fractions effectively:

• Identify common factors in both the numerator and the denominator that can be canceled out to simplify the fraction.
• Rewrite terms by factoring in order to spot these common elements more easily, as shown with the common \((7-3x)\) term.
• Apply division rules intuitively to recognize where you can simplify expressions' powers, resulting in a reduced form.

In the original problem, recognizing the quotient form \(\left(7-\frac{3}{2}x\right) \cdot (7-3x)^{-3/2}\) showcases a well-simplified algebraic fraction, obtained through careful observation and manipulation of common factors. This allows for easier computation or further manipulation of the expression.